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Before we learn about the Least Common Multiple, we need to understand what multiples are.
Multiple: A multiple of a number is what you get when you multiply that number by any whole number (1, 2, 3, 4, ...).
To find multiples of 3, we multiply 3 by different whole numbers:
3 × 1 = 3
3 × 2 = 6
3 × 3 = 9
3 × 4 = 12
3 × 5 = 15
So the multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, ...
Key Point: Multiples go on forever! We can always multiply by a bigger number.
Students sometimes confuse multiples and factors. Let's see the difference!
Factors are numbers that divide evenly into a number.
Factors of 12: 1, 2, 3, 4, 6, 12
Multiples are what you get when you multiply a number.
Multiples of 12: 12, 24, 36, 48, 60, ...
Factors → Divide → Get SMALLER numbers (usually)
Multiples → Multiply → Get BIGGER numbers
Think of it this way: Factors go down from the number, multiples go up from the number!
The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.
Least Common Multiple (LCM): The smallest number that is a multiple of all the given numbers.
It's the first number that appears in all the multiples lists!
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
Multiples of 6: 6, 12, 18, 24, 30, 36...
Common multiples: 12, 24, 36...
The LEAST common multiple is 12!
The LCM is super useful in many situations:
To add 1/4 + 1/6, we need a common denominator.
The LCM of 4 and 6 is 12, so we use 12 as our common denominator!
1/4 + 1/6 = 3/12 + 2/12 = 5/12
Bus A arrives every 6 minutes. Bus B arrives every 8 minutes.
When will they both arrive at the same time?
Answer: LCM(6, 8) = 24 minutes!
Two gears are turning. One completes a turn every 4 seconds, the other every 10 seconds.
When do they both return to the starting position together?
Answer: LCM(4, 10) = 20 seconds!
Steps:
LCM(3, 5) = 15
Problem 1: Find the LCM of 4 and 6
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
Answer: LCM(4, 6) = 12
Problem 2: Find the LCM of 6 and 8
Multiples of 6: 6, 12, 18, 24, 30, 36...
Multiples of 8: 8, 16, 24, 32, 40...
Answer: LCM(6, 8) = 24
Problem 3: Find the LCM of 2 and 9
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20...
Multiples of 9: 9, 18, 27, 36...
Answer: LCM(2, 9) = 18
Problem 4: Find the LCM of 5 and 7
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
Multiples of 7: 7, 14, 21, 28, 35, 42...
Note: 5 and 7 are relatively prime (no common factors except 1), so we multiply them!
Answer: LCM(5, 7) = 35
Sometimes one number is already a multiple of the other. This makes finding the LCM super easy!
Multiples of 5: 5, 10, 15, 20, 25, 30...
Multiples of 15: 15, 30, 45...
Notice that 15 appears right away in both lists!
LCM(5, 15) = 15
If one number is a multiple of the other, the larger number is the LCM!
More examples:
• LCM(3, 12) = 12 (because 12 is a multiple of 3)
• LCM(4, 20) = 20 (because 20 is a multiple of 4)
• LCM(6, 18) = 18 (because 18 is a multiple of 6)
Problem 5: Find the LCM of 7 and 21
Since 21 is a multiple of 7 (7 × 3 = 21), the larger number is the LCM.
Answer: LCM(7, 21) = 21
Problem 6: Find the LCM of 9 and 27
Since 27 is a multiple of 9 (9 × 3 = 27), the larger number is the LCM.
Answer: LCM(9, 27) = 27
Before we learn the prime factorization method for LCM, let's review what prime factorization is!
Prime Factorization: Breaking a number down into the prime numbers that multiply to make it.
Prime Number: A number that has only two factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13...
12
↙ ↘
2 × 6
↙ ↘
2 × 3
12 = 2 × 2 × 3 or 2² × 3
Steps:
Why this works: The LCM must contain all the prime factors needed to build each original number. By taking the highest power of each prime, we ensure the LCM is divisible by all the original numbers!
LCM(12, 18) = 36
LCM(8, 12) = 24
Problem 7: Find the LCM of 6 and 9 using prime factorization
6 = 2¹ × 3¹
9 = 3²
Highest power of 2: 2¹
Highest power of 3: 3²
LCM = 2¹ × 3² = 2 × 9 = 18
Answer: LCM(6, 9) = 18
Problem 8: Find the LCM of 10 and 15 using prime factorization
10 = 2¹ × 5¹
15 = 3¹ × 5¹
Highest power of 2: 2¹
Highest power of 3: 3¹
Highest power of 5: 5¹
LCM = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30
Answer: LCM(10, 15) = 30
Problem 9: Find the LCM of 14 and 21 using prime factorization
14 = 2¹ × 7¹
21 = 3¹ × 7¹
Highest power of 2: 2¹
Highest power of 3: 3¹
Highest power of 7: 7¹
LCM = 2¹ × 3¹ × 7¹ = 2 × 3 × 7 = 42
Answer: LCM(14, 21) = 42
Problem 10: Find the LCM of 20 and 30 using prime factorization
20 = 2² × 5¹
30 = 2¹ × 3¹ × 5¹
Highest power of 2: 2²
Highest power of 3: 3¹
Highest power of 5: 5¹
LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
Answer: LCM(20, 30) = 60
Sometimes two numbers have NO common factors except 1. These are called relatively prime numbers.
Relatively Prime: Two numbers that have no common factors except 1.
Examples: 7 and 9, 8 and 15, 11 and 20
If two numbers are relatively prime, their LCM is simply the product of the two numbers!
LCM = a × b
7 and 9 have no common factors (both are relatively prime).
7 = 7 (prime)
9 = 3²
LCM = 7 × 9 = 63
Problem 11: Find the LCM of 11 and 13
Both 11 and 13 are prime numbers, so they're automatically relatively prime!
LCM = 11 × 13 = 143
Answer: LCM(11, 13) = 143
Problem 12: Find the LCM of 8 and 15
8 = 2³ and 15 = 3 × 5
They share no common prime factors, so they're relatively prime!
LCM = 8 × 15 = 120
Answer: LCM(8, 15) = 120
There's an amazing connection between the Greatest Common Factor (GCF) and the Least Common Multiple (LCM)!
GCF(a, b) × LCM(a, b) = a × b
This means: If you know the GCF, you can find the LCM!
LCM(a, b) = (a × b) ÷ GCF(a, b)
Steps:
LCM = (a × b) ÷ GCF
LCM(8, 12) = 24
Problem 13: Find the LCM of 6 and 15 using the GCF method
Step 1: Find GCF(6, 15)
Factors of 6: 1, 2, 3, 6
Factors of 15: 1, 3, 5, 15
GCF(6, 15) = 3
Step 2: Use formula: LCM = (6 × 15) ÷ 3
LCM = 90 ÷ 3 = 30
Answer: LCM(6, 15) = 30
Problem 14: Find the LCM of 16 and 24 using the GCF method
Step 1: Find GCF(16, 24)
16 = 2⁴ and 24 = 2³ × 3
GCF(16, 24) = 2³ = 8
Step 2: Use formula: LCM = (16 × 24) ÷ 8
LCM = 384 ÷ 8 = 48
Answer: LCM(16, 24) = 48
What if we need to find the LCM of three numbers? No problem!
Strategy:
LCM(a, b, c) = LCM(LCM(a, b), c)
LCM(4, 6, 8) = 24
For three numbers, the prime factorization method is often easier!
LCM(6, 8, 9) = 72
Problem 15: Find the LCM of 3, 4, and 6
Method 1: Step by step
LCM(3, 4) = 12
LCM(12, 6) = 12 (since 12 is already a multiple of 6!)
Method 2: Prime factorization
3 = 3¹, 4 = 2², 6 = 2¹ × 3¹
LCM = 2² × 3¹ = 4 × 3 = 12
Answer: LCM(3, 4, 6) = 12
Problem 16: Find the LCM of 2, 5, and 10
Prime factorization:
2 = 2¹
5 = 5¹
10 = 2¹ × 5¹
Highest power of 2: 2¹
Highest power of 5: 5¹
LCM = 2¹ × 5¹ = 10
Note: 10 is already a multiple of both 2 and 5!
Answer: LCM(2, 5, 10) = 10
Problem 17: Find the LCM of 4, 5, and 6
Prime factorization:
4 = 2²
5 = 5¹
6 = 2¹ × 3¹
Highest power of 2: 2²
Highest power of 3: 3¹
Highest power of 5: 5¹
LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
Answer: LCM(4, 5, 6) = 60
Problem 18: Find the LCM of 6, 9, and 12
Prime factorization:
6 = 2¹ × 3¹
9 = 3²
12 = 2² × 3¹
Highest power of 2: 2²
Highest power of 3: 3²
LCM = 2² × 3² = 4 × 9 = 36
Answer: LCM(6, 9, 12) = 36
Wrong thinking: "LCM means finding the biggest factor."
Correct: LCM is about finding the smallest multiple, not a factor!
Remember: LCM is usually bigger than both original numbers (or equal to the largest one).
Wrong: LCM(6, 8) = 6 × 8 = 48
Correct: LCM(6, 8) = 24
Remember: Multiplying gives you a common multiple, but not always the least common multiple!
Wrong: When finding LCM(4, 6), listing only 4, 8 for multiples of 4.
Correct: Keep listing until you find a common multiple!
4, 8, 12 ← There it is!
Bus Route A comes to the stop every 12 minutes.
Bus Route B comes to the same stop every 18 minutes.
Question: If both buses arrive at 8:00 AM, when will they both arrive at the same time again?
Step 1: We need LCM(12, 18)
12 = 2² × 3¹
18 = 2¹ × 3²
Step 2: LCM = 2² × 3² = 4 × 9 = 36
Step 3: They'll meet again in 36 minutes
Answer: 8:36 AM
After that, they'll meet every 36 minutes: 9:12 AM, 9:48 AM, etc.
The main reason we learn LCM is to help us add and subtract fractions with different denominators!
Question: How much pizza did you eat together?
Step 1: Find LCM(4, 6) = 12
Step 2: Convert both fractions to have denominator 12
1/4 = 3/12 (multiply top and bottom by 3)
1/6 = 2/12 (multiply top and bottom by 2)
Step 3: Now we can add!
3/12 + 2/12 = 5/12
Answer: You ate 5/12 of the pizza together!
Key Point: The LCM becomes our common denominator!
Two traffic lights are side by side.
Light A turns green every 20 seconds.
Light B turns green every 30 seconds.
They both just turned green at the same time.
Question: How long until they're both green again at the same time?
Step 1: Find LCM(20, 30)
20 = 2² × 5¹
30 = 2¹ × 3¹ × 5¹
Step 2: LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
Answer: They'll both be green again in 60 seconds (1 minute)!
Light A will cycle 3 times (20 × 3 = 60)
Light B will cycle 2 times (30 × 2 = 60)
Listing Multiples - Best for:
✓ Small numbers (under 15)
✓ When you want to see all the common multiples
✓ Quick problems
Prime Factorization - Best for:
✓ Larger numbers
✓ Three or more numbers
✓ When listing would take too long
Using GCF - Best for:
✓ When you already know the GCF
✓ Two numbers only
✓ Checking your answer from another method
Use more than one method to check your answer!
If you get the same LCM using different methods, you know you're right!
1. What is LCM?
The smallest number that is a multiple of all given numbers.
2. Three Methods to Find LCM:
• Listing multiples (good for small numbers)
• Prime factorization (good for larger numbers)
• Using GCF formula: LCM = (a × b) ÷ GCF
3. Special Cases:
• When one number is a multiple of the other → LCM is the larger number
• When numbers are relatively prime → LCM = a × b
4. Why It Matters:
• Adding/subtracting fractions with different denominators
• Solving scheduling and pattern problems
• Finding when events happen together
Problem 19: Find the LCM of 15 and 25
Prime Factorization Method:
15 = 3¹ × 5¹
25 = 5²
LCM = 3¹ × 5² = 3 × 25 = 75
Answer: LCM(15, 25) = 75
Problem 20: Find the LCM of 18 and 24
Prime Factorization Method:
18 = 2¹ × 3²
24 = 2³ × 3¹
LCM = 2³ × 3² = 8 × 9 = 72
Check with GCF method:
GCF(18, 24) = 6
LCM = (18 × 24) ÷ 6 = 432 ÷ 6 = 72 ✓
Answer: LCM(18, 24) = 72
Challenge 1: Find the LCM of 12, 15, and 20
Prime Factorization:
12 = 2² × 3¹
15 = 3¹ × 5¹
20 = 2² × 5¹
Highest powers:
2² (from 12 or 20), 3¹ (from 12 or 15), 5¹ (from 15 or 20)
LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
Answer: LCM(12, 15, 20) = 60
Challenge 2: Three runners run around a track. Runner A completes a lap in 6 minutes, Runner B in 8 minutes, and Runner C in 9 minutes. If they all start together, when will they all be at the starting line again at the same time?
We need LCM(6, 8, 9)
6 = 2¹ × 3¹
8 = 2³
9 = 3²
LCM = 2³ × 3² = 8 × 9 = 72
Runner A: 72 ÷ 6 = 12 laps
Runner B: 72 ÷ 8 = 9 laps
Runner C: 72 ÷ 9 = 8 laps
Answer: 72 minutes (1 hour and 12 minutes)
Tip 1: LCM is always ≥ the largest number
Tip 2: If numbers share no factors (relatively prime), multiply them
Tip 3: If one number divides evenly into another, the larger is the LCM
Tip 4: For 2 and any even number, LCM = that even number
Tip 5: Check your answer: Can both original numbers divide evenly into your LCM?
LCM(7, 13) = ? → Both prime → 7 × 13 = 91
LCM(5, 25) = ? → 25 is multiple of 5 → 25
LCM(2, 14) = ? → 2 and even → 14
Try these in your head or on paper, then check your answers:
1. LCM(3, 7) = ?
2. LCM(4, 10) = ?
3. LCM(6, 9) = ?
4. LCM(5, 15) = ?
5. LCM(8, 12) = ?
1. LCM(3, 7) = 21 (relatively prime → multiply)
2. LCM(4, 10) = 20 (2² × 5 = 20)
3. LCM(6, 9) = 18 (2 × 3² = 18)
4. LCM(5, 15) = 15 (15 is multiple of 5)
5. LCM(8, 12) = 24 (2³ × 3 = 24)
What You Mastered Today:
✓ What multiples and LCM are
✓ Three different methods to find LCM
✓ Finding LCM of 2, 3, or more numbers
✓ Special shortcuts and patterns
✓ Real-world applications
✓ The connection between GCF and LCM
Coming Up Next:
Using LCM to add & subtract fractions
Comparing fractions with different denominators
More real-world problem solving
Keep practicing! You're doing amazing! 🌟